On a three-dimensional free boundary problem modeling electrostatic MEMS (Q345905)

Summary: We consider the dynamics of an electrostatically actuated thin elastic plate being clamped at its boundary above a rigid plate. While the existing literature focuses so far on a two-dimensional geometry, the present model considers a three-dimensional device where the harmonic electrostatic potential varies in the three-dimensional time-dependent region between the plates. The elastic plate deflection evolves according to a fourth-order semilinear parabolic equation which is coupled to the square of the gradient trace of the electrostatic potential on this plate. The strength of the coupling is tuned by a parameter \(\lambda\) proportional to the square of the applied voltage. We prove that this free boundary problem is locally well-posed in time and that for small values of \(\lambda\) solutions exist globally in time. We also derive the existence of a branch of asymptotically stable stationary solutions for small values of \(\lambda\) and non-existence of stationary solutions for large values thereof, the latter being restricted to a disc-shaped plate.